 Solution

Assume no induced graph on $P_i$ has the random-graph extension property. For each $i$, choose finite disjoint $A_i,B_i\subseteq P_i$ such that no vertex of $P_i\setminus (A_i\cup B_i)$ realizes the prescribed adjacency pattern. The unions $A={\bigcup }_i{A}_i$ and $B={\bigcup }_i{B}_i$ are finite and disjoint. The extension property in $M$ supplies a new vertex $v$ adjacent to all of $A$ and none of $B$. But $v\in P_j$ for some $j$, contradicting the choice of $A_j,B_j$. Thus some $P_j$ satisfies the extension property. It is a countable model of the theory of the random graph, so part (a) gives